Integrating Various Fields of Mathematics in the Process of Developing Multiple Solutions to the Same Problems in Geometry

Autor/inn/en	Stupel, Moshe; Oxman, Victor
Titel	Integrating Various Fields of Mathematics in the Process of Developing Multiple Solutions to the Same Problems in Geometry
Quelle	In: Australian Senior Mathematics Journal, 32 (2018) 1, S.26-41 (16 Seiten) PDF als Volltext Verfügbarkeit
Sprache	englisch
Dokumenttyp	gedruckt; online; Zeitschriftenaufsatz
ISSN	0819-4564
Schlagwörter	Mathematics Instruction; Geometry; Problem Solving; Mathematical Logic; Validity; Mathematical Concepts; Equations (Mathematics) + Suchen Sie Ihr Suchwort? Mathematics lessons; Mathematikunterricht; Geometrie; Problemlösen; Mathematical logics; Mathematische Logik; Gültigkeit; Equations; Mathematics; Gleichungslehre
Abstract	The solution of problems and the provision of proofs have always played a crucial part in mathematics. In fact, they are the heart and soul of this discipline. Moreover, the use of different techniques and methods of proof in the same mathematical field, or by combining fields, for the same specific problem, can show the interrelations between the fields, as well as the richness, beauty and elegance of mathematics. In addition to the specific roles of proof in mathematics, the authors suggest that attempts to also prove a certain result (or solve a problem) using methods from several other different areas of mathematics (geometry, trigonometry, analytic geometry, vectors, complex numbers, etc.) are very important in developing deeper mathematical understanding, creativity, and appreciation of the value of argumentation and proof in learning different topics of mathematics. Their approach, that of presenting multiple proofs to the same problem, as a device for constructing mathematical connections is supported by (Polya,1973,1981; Schoenfeld, 1988; NCTM, 2000; Ersoz, 2009; Levav-Waynberg & Leikin, 2009). Multiple solution tasks (MSTs) contain an explicit requirement for proving a statement in multiple ways. The differences between the proofs are based on using: (1) different representations of a mathematical concept; (2) different properties (definitions or theorems) of mathematical concepts from a particular mathematical topic; (3) different mathematics tools and theorems from different branches of mathematics; (4) different tools and theorems from different subjects (not necessarily mathematics); and (5) different strategies of problem solving. In this case, the authors apply the third type of differences between the proofs. In a student activity, they present various solutions to the problems using the tools and theorems of Euclidean geometry, analytic geometry, trigonometry, and vectors. (ERIC).
Anmerkungen	Australian Association of Mathematics Teachers (AAMT). GPO Box 1729, Adelaide 5001, South Australia. Tel: +61-8-8363-0288; Fax: +61-8-8362-9288; e-mail: office@aamt.edu.au; Web site: http://www.aamt.edu.au
Erfasst von	ERIC (Education Resources Information Center), Washington, DC
Update	2021/2/06